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CategoriesFunctorsTransformations.png?

Duality is of course an ancient notion in philosophy. At least as a term, it makes a curious re-appearance in the conjectural theory of fundamental physics formerly known as string theory, in the guise of duality in string theory. In both cases, the literature left some room in delineating what precisely is meant. But the philosophically inclined mathematician could notice (see Lambek 82) that an excellent candidate to make precise the idea of duality is the mathematical concept of adjunction, from category theory. This is particularly pronounced for adjoint triples (Remark below) and their induced adjoint modalities (Lawvere 91, which exhibit a given “mode of being” of any object $X$ as intermediate between two dual opposite extremes:

$$\Box X \overset{\phantom{AAAA}}{\longrightarrow} X \overset{\phantom{AAAA}}{\longrightarrow} \bigcirc X$$

For example, cohesive geometric structure on generalized spaces is captured, this way, as modality in between the discrete and the codiscrete.

Historically, category theory was introduced in order to make precise the concept of natural transformation: The concept of functors was introduced just so as to support that of natural transformations, and the concept of categories only served that of functors (see Freyd 1964 p 1).

But natural transformations are, in turn, exactly the basis for the concept of adjoint functors (Def. below), equivalently adjunctions between categories (Prop. below), shown on the left. All universal constructions — the heart of category theory — are special cases of adjoint functors, hence of dualities, if we follow Lambek 82: This includes the concepts of limits and colimits, ends and coends, Kan extensions (Prop. below), and the behaviour of these constructions, such as for instance the free co-completion nature of the Yoneda embedding.

• Maissam Barkeshli, Xiao-Liang Qi: Synthetic Topological Qubits in Conventional Bilayer Quantum Hall Systems, Phys. Rev. X 4 (2014) 041035 [doi:10.1103/PhysRevX.4.041035, arXiv:1302.2673]

• Roger S. K. Mong et al.: Universal Topological Quantum Computation from a Superconductor-Abelian Quantum Hall Heterostructure, Phys. Rev. X 4 (2014) 011036 [doi:10.1103/PhysRevX.4.011036, arXiv:1307.4403]

• D. V. Averin, V. J. Goldman: Quantum computation with quasiparticles of the Fractional Quantum Hall Effect, Solid State Communications 121 1 (2001) 25-28 [doi:10.1016/S0038-1098(01)00447-1, arXiv:cond-mat/0110193]

• A. P. Balachandran, A. Ibort, G. Marmo, M. Martone: Quantum Geons and Noncommutative Spacetimes

$$0 \to \pi_1(Maps) \to \pi_2\big(Maps \sslash Diff\big) \to B (\mathbb{Z} \times \mathbb{Z}) \to \pi_1(Maps) \to \pi_1\big(Maps \sslash Diff\big) \to B MCG \to \mathbb{Z} \to \mathbb{Z} \to \pi_0(B Diff)$$

Annular braid group

The annular braid group $Br_n(An)$ on $n \in \mathbb{N}$ strands is the surface braid group of the annulus, hence the fundamental group of the [configuration space of points|configuration space of -points]] inside the annulus.

Properties

Proposition

The annular braid group is isomorphic to a semidirect product

$$Br_n(An) \,\simeq\, Br_n^{aff} \rtimes \mathbb{Z}$$

of the the affine braid group? with the group of integers, the latter generated by the braid which exhibits a 1-step cyclic permutation.

References

• Richard P. Kent, David Pfeifer: A Geometric and algebraic description a Of annular braid groups, International Journal of Algebra and Computation 12 01n02 (2002) 85-97 [doi:10.1142/S0218196702000997]

• Paolo Bellingeri, Arnaud Bodin: The braid group of a necklace, Mathematische Zeitschrift 283 3 (2014) [arXiv:1404.2511, doi:10.1007/s00209-016-1630-0]

Asymptotic gauge groups

Given a gauge theory (and/or gravity) on a spacetime with asymptotic boundary, certain would-be gauge transformations (diffeomorphisms) that act non-trivially on asymptotic “boundary data” may in fact be identified as physically observable global symmetries and hence have, in contrast to actual gauge symmetries, “direct empirical significance” (DES, Teh 2016).

A key example is (symmetry generated by) the ADM mass and generally the BMS group? of asymptotic symmetries in asymptotically flat spacetimes.

Up to technical fine-print (cf. Borsboom & Posthuma 2015) a group of asymptotic symmetries is the coset space of all gauge symmetries that respect boundary data by the subgroup of bulk gauge transformations which act as the identity map on the asymptotic boundary (cf. Strominger 2018 (2.10.1), Borsboom & Posthuma 2015 p 2):

$$AsymptoticSymmetries \;=\; \frac{ BoundaryAdmissibleGaugeSymmetries }{ BulkGaugeSymmetriesTrivialAtBoundary }$$

Hence if the bulk gauge symmetries form a normal subgroup then the asymptotic symmetries form a quotient group characterized by a short exact sequence of the form

$$1 \to BulkGaugeSymmetriesTrivialAtBoundary \longrightarrow BoundaryAdmissibleGaugeSymmetries \longrightarrow AsymptoticSymmetries \to 1 \,.$$

References

• Steve Carlip: Dynamics of Asymptotic Diffeomorphisms in $(2+1)$-Dimensional Gravity, Class. Quant. Grav. 22 (2005) 3055-3060 [arXiv:gr-qc/0501033, doi:10.1088/0264-9381/22/14/014]

• Nicholas Teh: Galileo’s Gauge: Understanding the Empirical Significance of Gauge Symmetry, Philosophy of Science 83 (2016) 93-118.

• Andrew Strominger: Asymptotic Symmetris, section 2.10 in: Lectures on the Infrared Structure of Gravity and Gauge Theory, Princeton University Press (2018) [ISBN:9780691179506,

  arXiv:1703.05448]

• Silvester Borsboom, Hessel Posthuma: Global Gauge Symmetries and Spatial Asymptotic Boundary Conditions in Yang-Mills theory [arXiv:2502.16151]

pdf

• J. Tolar: On Clifford groups in quantum computing [arXiv:1810.10259]

Definition

By the spherical braid group $Br_n(S^2)$, for $n \in \mathbb{N}$, one means the surface braid group

$$Br_n(S^2) \;\simeq\; \pi_1 Conf_n(S^2) \,,$$

where the surface in question is the 2-sphere $S^2$. Hence the surface braid group is the fundamental group $\pi_1(-)$ of the configuration space of $n$-points, $Conf_n(-)$, on the 2-sphere.

Properties

(from math.SE:q/136821)

Proposition

The spherical braid group is the quotient group of the ordinary braid group by one further relation:

$$Br_n(S^2) \;\simeq\; Br_n/ \big( (b_1 b_2 \cdots b_{n-1})(b_{n-1} \cdots b_2 b_1) \,,$$

where the $b_i$ denote the Artin braid generators.

Moreover, the canonical map from the plain braid group to the symmetric group factors through this quotient map to the spherical braid group

(Fadell & Van Buskirk 1961 p 245, 255, cf. Tan 2024 §3.1)

References

• Edward Fadell, James Van Buskirk: On the braid groups of $E^2$ and $S^2$, Bull. Amer. Math. Soc. 67 2 (1961) 211-213 [euclid:bams/1183524083]

• Cindy Tan: Smallest nonabelian quotients of surface braid groups, Algebr. Geom. Topol. 24 (2024) 3997-4006 [arXiv:2301.01872, doi:10.2140/agt.2024.24.3997]

$$\pi_2 S^2 \to \pi_1 \Maps^\ast(S^2, S^2) \to \pi_1 Maps(S^2, S^2) \to \pi_1 S^2 \to \pi_0 \Maps^\ast(S^2, S^2) \to \pi_0 Maps(S^2, S^2) \to \pi_0 S^2$$

Example

The unitarization of the standard representation of the symmetric group $Sym_3$ has the two generating transpositions represented by (what in quantum information theory is called)

1. the Pauli Z-gate $Z$

2. its composition $- Z \circ R_y(3\pi/2)$ with rotation gates.

Proof

For definiteness of computation, when group averaging we will be cycling through the elements of $Sym_3$ in this order:

$$Sym_3 \;=\; \left\{ \begin{array}{l} (1 2 3) ,\,\\ (1 3 2) ,\,\\ (2 1 3) ,\,\\ (2 3 1) ,\,\\ (3 1 2) ,\,\\ (3 2 1) \end{array} \right\} \,.$$

Let $\big\{e_1, e_2, e_3\big\}$ denote the canonical linear basis of the defining representation, with group action given by

$$\sigma \cdot e_i \;\coloneqq\; e_{\sigma(i)} \,.$$

Then a linear basis for the standard representation inside the defining representation is:

$$\begin{array}{ccc} \left[\begin{array}{c}1 \\ 0\end{array}\right] \;\coloneqq\; e_1 - e_3 \\ \left[\begin{array}{c}0 \\ 1\end{array}\right] \;\coloneqq\; e_2 - e_3 \mathrlap{\,.} \end{array}$$

The group-averaged inner product of these basis elements is found to be:

$$\left\langle \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] ,\, \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \right\rangle \;=\; \left( \;\; \begin{array}{l} \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] + \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] + \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \end{array} \right)/6 \;=\; 4/6 \;=\; 2/3$$

$$\left\langle \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] ,\, \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \right\rangle \;=\; \left( \;\; \begin{array}{l} \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] + \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] + \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \end{array} \right)/6 \;=\; 8/6 \;=\; 4/3$$

$$\left\langle \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] ,\, \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \right\rangle \;=\; \left( \;\; \begin{array}{l} \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] + \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] + \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] + \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] \cdot \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] + \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] \cdot \left[ \begin{array}{c} -1 \\ 0 \end{array} \right] \end{array} \right)/6 \;=\; 8/6 \;=\; 4/3$$

From this an orthonormal basis for the averaged inner product is

$$\begin{array}{l} {\vert 0 \rangle} \;\coloneqq\; \tfrac{1}{2} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \;=\; \tfrac{1}{2}(e_1 + e_2) - e_3 \\ {\vert 1 \rangle} \;\coloneqq\; \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2}(e_1 - e_2) \,. \end{array}$$

On this bases the action of $(213)$ is

$$\rho(213)\big({\vert 0 \rangle}\big) \;=\; {\vert 0 \rangle}$$

$$\rho(213)\big({\vert 1 \rangle}\big) \;=\; -{\vert 1 \rangle}$$

and hence $(213)$ acts as the Pauli Z-gate:

$$\rho(213) \;=\; \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \;=\; Z$$

On the other hand, the action of $(132)$ is found to be

$$\rho(132)\big({\vert 0 \rangle}\big) \;=\; \rho(132) \left( \tfrac{1}{2} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \right) \;=\; \tfrac{1}{2} \left[ \begin{array}{c} 1 \\ -2 \end{array} \right] \;=\; -\tfrac{1}{2} {\vert 0 \rangle} + \tfrac{\sqrt{3}}{2} {\vert 1 \rangle}$$

$$\rho(132)\big({\vert 1 \rangle}\big) \;=\; \rho(132) \left( \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \right) \;=\; \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2} {\vert 0 \rangle} + \tfrac{1}{2} {\vert 1 \rangle}$$

and hence

$$\rho(132) \;=\; \left( \begin{array}{c} -1/2 & \sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{array} \right) \;=\; \left( \begin{array}{c} -1 & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array}{c} 1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{array} \right) \;=\; \left( \begin{array}{c} -1 & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array}{c} cos(\pi/3) & -sin(\pi/3) \\ sin(\pi/3) & cos(\pi/3) \end{array} \right) \;=\; - Z \circ R_y(2\pi/3)$$

end

\lnebreak

The action of $(231)$

$$(231) {\vert 0 \rangle} \;=\; \tfrac{1}{2} \left[ \begin{array}{c} -2 \\ 1 \end{array} \right] \;=\; -\tfrac{1}{2} {\vert 0 \rangle} -\tfrac{\sqrt{3}}{2} {\vert 1 \rangle}$$

$$(231) {\vert 1 \rangle} \;=\; \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2} {\vert 0 \rangle} - \tfrac{1}{2} {\vert 1 \rangle}$$

Hence

$$(231) \;=\; \left[ \begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & - 1/2 \end{array} \right]$$

The action of $(321)$

$$(321) {\vert 0 \rangle} \;=\; (321) \tfrac{1}{2} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \;=\; \tfrac{1}{2} \left[ \begin{array}{c} -2 \\ 1 \end{array} \right] \;=\; -\tfrac{1}{2} {\vert 0 \rangle} - \tfrac{\sqrt{3}}{2} {\vert 1 \rangle}$$

$$(321) {\vert 1 \rangle} \;=\; (321) \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2} \left[ \begin{array}{c} 0 \\ -1 \end{array} \right] \;=\; \tfrac{\sqrt{3}}{2} {\vert 0 \rangle} - \tfrac{1}{2} {\vert 1 \rangle}$$

Hence $(321)$ acts as

$$(321) \;=\; \left[ \begin{array}{c} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & -1/2 \end{array} \right] \;=\; - \left[ \begin{array}{c} 1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{array} \right] \;=\; - R_y(2\pi/3)$$

Defect anyons

• Maissam Barkeshli, Chao-Ming Jian, Xiao-Liang Qi: Twist defects and projective non-Abelian braiding statistics, Phys. Rev. B 87 (2013) 045130 [doi:10.1103/PhysRevB.87.045130, arXiv:1208.4834]

Controlling anyons

• G. Rosenberg, B. Seradjeh, C. Weeks, M. Franz: Creation and manipulation of anyons in a layered superconductor-2DEG system, Phys. Rev. B 79 205102 (2009) [doi:10.1103/PhysRevB.79.205102, arXiv:0812.3140]

• Gábor B. Halász: Gate-Controlled Anyon Generation and Detection in Kitaev Spin Liquids, Phys. Rev. Lett. 132 206501 (2024) [doi:10.1103/PhysRevLett.132.206501]

Loop space of the 2-sphere

On the loop space $\Omega S^2$ of the 2-sphere

in relation to braid groups

• Frederick R. Cohen, J. Wu: On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere, in: Categorical Decomposition Techniques in Algebraic Topology, in Progress in Mathematics 215, Birkhäuser (2003) 93-105 [doi:10.1007/978-3-0348-7863-0_6]

On $\Omega S^2 \,\simeq\, B \Omega^2 S^2$ regarded as a classifying space (for “$\mathbf{l}$ine” bundles):

• Jack Morava: A homotopy-theoretic context for CKM/Birkhoff renormalization [arXiv:2307.10148, spire:2678618]

• Jack Morava: Some very low-dimensional algebraic topology [arXiv:2411.15885]